warm up - city university of new york

Warm up

Recall, to graph a conic function, you want it in the formparabola: (x− x0)2 = 4p(y − y0) or (y − y0)2 = 4p(x− x0),

ellipse:(x−x0a

)2+(y−y0

b

)2= 1,

hyperbola:(x−x0a

)2 − (y−y0b

)2= 1 or

(y−y0a

)2 − (x−x0b

)2= 1

Without doing any algebraic manipulation, decide whether each ofthe following conic sections is a parabola, ellipse, or hyperbola.Can you tell which way it’s oriented?

1. 36y2 − x2 + 72y + 6x+ 31 = 0

(x−32

)2 − (y+11/3

)2= 1

2. y2 − 10y − 8x+ 49 = 0

(y − 5)2 = 4(2)(x− 3)

3. y2 + 2x2 + 4y + 8x+ 11 = 0

(y + 2)2 +(

x+21/√2

)2= 1

Now, complete squares and put into a form you can graph from.Identify (as applicable) the vertices, foci, axes, directrixes,asymptotes.

Warm up

Recall, to graph a conic function, you want it in the formparabola: (x− x0)2 = 4p(y − y0) or (y − y0)2 = 4p(x− x0),

ellipse:(x−x0a

)2+(y−y0

b

)2= 1,

hyperbola:(x−x0a

)2 − (y−y0b

)2= 1 or

(y−y0a

)2 − (x−x0b

)2= 1

Without doing any algebraic manipulation, decide whether each ofthe following conic sections is a parabola, ellipse, or hyperbola.Can you tell which way it’s oriented?

1. 36y2 − x2 + 72y + 6x+ 31 = 0(x−32

)2 − (y+11/3

)2= 1

2. y2 − 10y − 8x+ 49 = 0 (y − 5)2 = 4(2)(x− 3)

3. y2 + 2x2 + 4y + 8x+ 11 = 0 (y + 2)2 +(

x+21/√2

)2= 1

Now, complete squares and put into a form you can graph from.Identify (as applicable) the vertices, foci, axes, directrixes,asymptotes.

36y2 − x2 + 72y + 6x+ 31 = 0: Hyperbola!

0 = 36y2−x2 + 72y + 6x+ 31

= 36(y2 + 2y)− (x2 − 6x) + 31

= 36((y + 1)2 − 1)− ((x− 3)2 − 9) + 31

= 36(y + 1)2 − (x− 3)2 + 4.

So 4 = (x− 3)2 − 36(y + 1)2. So

1 = (x− 3)2/4− 9(y + 1)2 =(x−32

)2 − (y+11/3

)2.

center: (3,−1); a = 2, b = 1/3,c =

√22 + (1/3)2 =

√37/3 ≈ 2.03; ;

vertices: (3± 2,−1);foci: (3± c,−1); (note: very close to the vertices!)

asymptotes: y + 1 = ±(1/32

)(x− 3)

5 10

-2

-1

y2 − 10y−8x+ 49 = 0: Sideways parabola! (opens to the right)

0 = y2−10y−8x+49 = (y−5)2−25−8x+49 = (y−5)2−8x+24,

so(y − 5)2 = 8x− 24 = 8(x− 3) = 4(2)(x− 3).

vertex: (3, 5); p = 2; focus: (3 + p, 5); directrix: x = 3− p

5

5

10

y2 + 2x2 + 4y + 8x+ 11 = 0: Ellipse!

0 = y2 + 2x2 + 4y + 8x+ 11

= (y2 + 4y) + 2(x2 + 4x) + 11

= ((y + 2)2 − 4) + 2((x+ 2)2 − 4) + 11

= (y + 2)2 + 2(x+ 2)2 − 1.

So

1 = (y + 2)2 + 2(x+ 2)2 = (y + 2)2 +

(x+ 2

1/√2

)2

.

center: (−2,−2);a = 1, b = 1/

√2,

c =√

12 − (1/√2)2 = 1/

√2 ≈ 0.71;

vertices: (−2,−2± a);foci: (−2,−2± c);major axis: x = −2

-3 -2 -1

-3

-2

-1

y2 + 2x2 + 4y + 8x+ 11 = 0: Ellipse!

0 = y2 + 2x2 + 4y + 8x+ 11

= (y2 + 4y) + 2(x2 + 4x) + 11

= ((y + 2)2 − 4) + 2((x+ 2)2 − 4) + 11

= (y + 2)2 + 2(x+ 2)2 − 1.

So

1 = (y + 2)2 + 2(x+ 2)2 = (y + 2)2 +

(x+ 2

1/√2

)2

.

center: (−2,−2);a = 1, b = 1/

√2,

c =√12 − (1/

√2)2 = 1/

√2 ≈ 0.71;

vertices: (−2,−2± a);foci: (−2,−2± c);major axis: x = −2

-3 -2 -1

-3

-2

-1

Conics with cross-terms

Again, all conic sections satisfy an equation of the form

Ax2 +Bxy + Cy2 +Dx+ Ey + F = 0

for some constants A,B,C,D,E, F .

Last time, we talked about how to analyze anything where B = 0.

Just like we can shift up/down or left/right to get from somethinglike

(y − 5)2 = 8(x− 3) to (y)2 = 8x,

we can rotate to get from something where B 6= 0 to somethingwhere B = 0.

Take the usual x, y-axis

, and rotate it by θ in [0, π/2)to get a new x, y-axis:

x

y

x

y

θ

(x, y)

r φ

We want to take a point (x, y) and write it in terms of x and y.Easier to do in polar!In the x, y-frame,

x = r cos(φ) y = r sin(φ).But in the x, y-frame, the angle is φ+ θ, and the radius is thesame! So in the x, y-frame,

x = r cos(φ+ θ) y = r sin(φ+ θ).Now, using the angle addition formulas,

x = r cos(φ+ θ) = r cos(φ) cos(θ)− r sin(φ) sin(θ) = x cos(θ)− y sin(θ),y = r sin(φ+ θ) = r cos(φ) sin(θ) + r sin(φ) cos(θ) = x sin(θ) + y cos(θ).

Combining these equations, we getx cos(θ) + y sin(θ) = x(cos2(θ) + sin2(θ)) + y · 0 = x, and−x sin(θ) + y cos(θ) = x · 0 + y(sin2(θ) + cos2(θ)) = y.

Take the usual x, y-axis, and rotate it by θ in [0, π/2)to get a new x, y-axis:

x

y

x

y

θ

(x, y)

r φ







x

y

x

y

θ

(x, y)

r φ

We want to take a point (x, y) and write it in terms of x and y.

Easier to do in polar!In the x, y-frame,






x

y

x

y

θ

(x, y)

r φ

We want to take a point (x, y) and write it in terms of x and y.Easier to do in polar!

In the x, y-frame,x = r cos(φ) y = r sin(φ).

But in the x, y-frame, the angle is φ+ θ, and the radius is thesame! So in the x, y-frame,





x

y

x

y

θ

(x, y)

r φ


x = r cos(φ) y = r sin(φ).

But in the x, y-frame, the angle is φ+ θ, and the radius is thesame! So in the x, y-frame,





x

y

x

y

θ

(x, y)

r φ


x = r cos(φ) y = r sin(φ).But in the x, y-frame, the angle is φ+ θ, and the radius is thesame!

So in the x, y-frame,x = r cos(φ+ θ) y = r sin(φ+ θ).

Now, using the angle addition formulas,x = r cos(φ+ θ) = r cos(φ) cos(θ)− r sin(φ) sin(θ) = x cos(θ)− y sin(θ),y = r sin(φ+ θ) = r cos(φ) sin(θ) + r sin(φ) cos(θ) = x sin(θ) + y cos(θ).



x

y

x

y

θ

(x, y)

r φ



x = r cos(φ+ θ) y = r sin(φ+ θ).




x

y

x

y

θ

(x, y)

r φ

x = r cos(φ),y = r sin(φ)

x = r cos(φ+ θ)y = r sin(φ+ θ)

Now, using the angle addition formulas,x = r cos(φ+ θ) = r cos(φ) cos(θ)− r sin(φ) sin(θ)

= x cos(θ)− y sin(θ),y = r sin(φ+ θ) = r cos(φ) sin(θ) + r sin(φ) cos(θ) = x sin(θ) + y cos(θ).



x

y

x

y

θ

(x, y)

r φ



Now, using the angle addition formulas,x = r cos(φ+ θ) = r cos(φ) cos(θ)− r sin(φ) sin(θ) = x cos(θ)− y sin(θ),

y = r sin(φ+ θ) = r cos(φ) sin(θ) + r sin(φ) cos(θ) = x sin(θ) + y cos(θ).Combining these equations, we get

x cos(θ) + y sin(θ) = x(cos2(θ) + sin2(θ)) + y · 0 = x, and−x sin(θ) + y cos(θ) = x · 0 + y(sin2(θ) + cos2(θ)) = y.


x

y

x

y

θ

(x, y)

r φ



Now, using the angle addition formulas,x = r cos(φ+ θ) = r cos(φ) cos(θ)− r sin(φ) sin(θ) = x cos(θ)− y sin(θ),y = r sin(φ+ θ) = r cos(φ) sin(θ) + r sin(φ) cos(θ)

= x sin(θ) + y cos(θ).Combining these equations, we get

x cos(θ) + y sin(θ) = x(cos2(θ) + sin2(θ)) + y · 0 = x, and−x sin(θ) + y cos(θ) = x · 0 + y(sin2(θ) + cos2(θ)) = y.


x

y

x

y

θ

(x, y)

r φ






x

y

x

y

θ

(x, y)

r φ




Combining these equations, we getx cos(θ) + y sin(θ) = x(cos2(θ) + sin2(θ)) + y · 0

= x, and−x sin(θ) + y cos(θ) = x · 0 + y(sin2(θ) + cos2(θ)) = y.


x

y

x

y

θ

(x, y)

r φ




Combining these equations, we getx cos(θ) + y sin(θ) = x(cos2(θ) + sin2(θ)) + y · 0 = x, and−x sin(θ) + y cos(θ) = x · 0 + y(sin2(θ) + cos2(θ))

= y.


x

y

x

y

θ

(x, y)

r φ





Change of coordinates for rotating axes:

x

y

x

y

θ

(x, y) = (x, y)

If I rotate the x, y-axis by θ to get new coordinates (x, y), then theconversion between coordinate systems is given by

x = x cos(θ)− y sin(θ) and y = x sin(θ) + y cos(θ)

x = x cos(θ) + y sin(θ) and y = −x sin(θ) + y cos(θ)

Example

Show that the function xy = 1 is a hyperbola rotated by π/4.What are the foci and the asymptotes?

Solution. Convert coordinates, using θ = π/4:

x = x cos(π/4)− y sin(π/4) =√22 (x− y), and

y = x sin(π/4) + y cos(π/4) =√22 (x+ y).

So, subbing into xy = 1, we get

1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.

In the x, y frame, the asymptotes are y = ±x. Buty = x is equivalent to x− y = 0, which is equivalent to x = 0, and

y = −x is equivalent to x+ y = 0, is equivalent to y = 0.Also, in the x, y frame, the foci are (x, y) = (±2, 0). So

x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example



x = x cos(π/4)− y sin(π/4)

=√22 (x− y), and

y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example



x = x cos(π/4)− y sin(π/4) =√22 (x− y)

, and

y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4)

=√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy

=(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)

= 12(x

2−y2) =(

x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2)

=(

x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.

In the x, y frame, the asymptotes are y = ±x.

Buty = x is equivalent to x− y = 0, which is equivalent to x = 0, and


x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.

In the x, y frame, the asymptotes are y = ±x. Buty = x is equivalent to x− y = 0

, which is equivalent to x = 0, andy = −x is equivalent to x+ y = 0, is equivalent to y = 0.

Also, in the x, y frame, the foci are (x, y) = (±2, 0). Sox = ±2 cos(π/4) + 0 · sin(π/4) = ±

√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.

In the x, y frame, the asymptotes are y = ±x. Buty = x is equivalent to x− y = 0, which is equivalent to x = 0

, andy = −x is equivalent to x+ y = 0, is equivalent to y = 0.

Also, in the x, y frame, the foci are (x, y) = (±2, 0). Sox = ±2 cos(π/4) + 0 · sin(π/4) = ±

√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.


y = −x is equivalent to x+ y = 0

, is equivalent to y = 0.Also, in the x, y frame, the foci are (x, y) = (±2, 0). So

x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.


y = −x is equivalent to x+ y = 0, is equivalent to y = 0.Also, in the x, y frame, the foci are (x, y) = (±2, 0).

Sox = ±2 cos(π/4) + 0 · sin(π/4) = ±

√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4)

= ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2

, andy = ±(2) sin(π/4) + 0 · cos(π/4) = ±

√2.

Example




y = x sin(π/4) + y cos(π/4) =√22 (x+ y).


1 = xy =(√

22 (x− y)

)(√22 (x+ y)

)= 1

2(x2−y2) =

(x√2

)2−(

y√2

)2.



x = ±2 cos(π/4) + 0 · sin(π/4) = ±√2, and

y = ±(2) sin(π/4) + 0 · cos(π/4) = ±√2.

Graphing xy = 1:

In summary, the function xy = 1 is the function(

x√2

)2−(

y√2

)2rotated by θ = π/4. In the x, y-frame, the asymptotes are x = 0and y = 0. The foci are (

√2,√2) and (−

√2,−√2).

^𝑥

^𝑦

𝑥

𝑦

(Compare to y = 1/x!)

Hypothesis: for a conic sectionAx2 +Bxy + Cy2 +Dx+ Ey + F = 0, there is some rotationalchange of coordinates in which the xy term is missing.

In the previous example, we were explicitly given the right angle torotate by to eliminate the xy-term.

Big question: How could we find this in general?

Strategy:Take a generic θ and plug the change of coordinates

x = x cos(θ)− y sin(θ), y = x sin(θ) + y cos(θ)

in to the generic conic section equation.Collect “like terms” in x and y.The coefficient of xy will be in terms of θ, A, B, etc.. Solve for θso that, in this new equation, the coefficient of xy is 0.






in to the generic conic section equation.

Collect “like terms” in x and y.The coefficient of xy will be in terms of θ, A, B, etc.. Solve for θso that, in this new equation, the coefficient of xy is 0.






in to the generic conic section equation.Collect “like terms” in x and y.

The coefficient of xy will be in terms of θ, A, B, etc.. Solve for θso that, in this new equation, the coefficient of xy is 0.






in to the generic conic section equation.Collect “like terms” in x and y.The coefficient of xy will be in terms of θ, A, B, etc.. Solve for θso that, in this new equation, the coefficient of xy is 0.


0 = Ax2 +Bxy + Cy2 +Dx+ Ey + F

= A(x cos(θ)− y sin(θ))(x cos(θ)− y sin(θ))+B(x cos(θ)− y sin(θ))(x sin(θ) + y cos(θ))

+ C(x sin(θ) + y cos(θ))(x sin(θ) + y cos(θ))

+D(x cos(θ)− y sin(θ)) + E(x sin(θ) + y cos(θ)) + F

Coefficient of xy:−A cos(θ) sin(θ)−A sin(θ) cos(θ)

+B cos(θ) cos(θ)−B sin(θ) sin(θ)+C sin(θ) cos(θ) + C cos(θ) sin(θ)

= B(cos2(θ)− sin2(θ)) + (C −A)(2 cos(θ) sin(θ))= B cos(2θ) + (C −A) sin(2θ).

So we want θ such that

B cos(2θ) + (C −A) sin(2θ) = 0, i.e. tan(2θ) = B/(A− C) .






Coefficient of xy:

−A cos(θ) sin(θ)−A sin(θ) cos(θ)+B cos(θ) cos(θ)−B sin(θ) sin(θ)+C sin(θ) cos(θ) + C cos(θ) sin(θ)









Coefficient of xy:−A cos(θ) sin(θ)

−A sin(θ) cos(θ)+B cos(θ) cos(θ)−B sin(θ) sin(θ)+C sin(θ) cos(θ) + C cos(θ) sin(θ)










+B cos(θ) cos(θ)

−B sin(θ) sin(θ)+C sin(θ) cos(θ) + C cos(θ) sin(θ)










+B cos(θ) cos(θ)−B sin(θ) sin(θ)

+C sin(θ) cos(θ) + C cos(θ) sin(θ)= B(cos2(θ)− sin2(θ)) + (C −A)(2 cos(θ) sin(θ))= B cos(2θ) + (C −A) sin(2θ).









+B cos(θ) cos(θ)−B sin(θ) sin(θ)+C sin(θ) cos(θ)

+ C cos(θ) sin(θ)= B(cos2(θ)− sin2(θ)) + (C −A)(2 cos(θ) sin(θ))= B cos(2θ) + (C −A) sin(2θ).










= B(cos2(θ)− sin2(θ)) + (C −A)(2 cos(θ) sin(θ))

= B cos(2θ) + (C −A) sin(2θ).So we want θ such that











B cos(2θ) + (C −A) sin(2θ) = 0

, i.e. tan(2θ) = B/(A− C) .



Rotating by θ eliminates the xy terms whentan(2θ) = B/(A− C).

What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?

Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.

Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4

, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1.

So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.

You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0




What θ do you want to rotate by to graph each of the followingconic sections?Example: 8x2 − 4xy + 4y2 + 1 = 0.Solution: A = 8, B = −4, C = 4, so tan(2θ) = 4/(8− 4) = 1. So2θ = π/4. So θ = π/8.You try:

1. x2 − xy + y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0


x = x cos(θ)−y sin(θ), y = x sin(θ)+y cos(θ), tan(2θ) = B/(A−C).

Of course, plugging in things like θ = π/8 into sin(θ) and cos(θ)isn’t fun without a calculator. Draw a triangle for cos(2θ) and usethe half angle formulas!

cos2(θ) = 12(1 + cos(2θ)) sin2(θ) = 1

2(1− cos(2θ))

Example: If tan(2θ) = 1, then 2θ is the angle of the triangle

1

1c

2θ

c =√12 + 12 =

√2

cos(2θ) = 1/√2 =√2/2

cos(θ) =√

12(1 +

√2/2)

sin(θ) =√

12(1−

√2/2)

So

x = x√

12 (1 +

√22 )−y

√12 (1−

√22 ), y = x

√12 (1−

√22 )+y

√12 (1 +

√22 )



Of course, plugging in things like θ = π/8 into sin(θ) and cos(θ)isn’t fun without a calculator.

Draw a triangle for cos(2θ) and usethe half angle formulas!

cos2(θ) = 12(1 + cos(2θ)) sin2(θ) = 1

2(1− cos(2θ))


1

1c

2θ

c =√12 + 12 =

√2

cos(2θ) = 1/√2 =√2/2

cos(θ) =√

12(1 +

√2/2)

sin(θ) =√

12(1−

√2/2)

So

x = x√

12 (1 +

√22 )−y

√12 (1−

√22 ), y = x

√12 (1−

√22 )+y

√12 (1 +

√22 )



Of course, plugging in things like θ = π/8 into sin(θ) and cos(θ)isn’t fun without a calculator. Draw a triangle for cos(2θ) and usethe half angle formulas!

cos2(θ) = 12(1 + cos(2θ)) sin2(θ) = 1

2(1− cos(2θ))


1

1c

2θ

c =√12 + 12 =

√2

cos(2θ) = 1/√2 =√2/2

cos(θ) =√

12(1 +

√2/2)

sin(θ) =√

12(1−

√2/2)

So

x = x√

12 (1 +

√22 )−y

√12 (1−

√22 ), y = x

√12 (1−

√22 )+y

√12 (1 +

√22 )



tan(2θ) = B/(A− C).cos2(θ) = 1

2(1 + cos(2θ)) sin2(θ) = 12(1− cos(2θ))

You try: Find the change of coordinates that will eliminate the xyterm in each of the following conic sections.

1. x2 − xy − y2 − 2 = 02. 4x2 − 4xy + 7y2 − 24 = 03. 2x2 + 4

√3xy + 6y2 − 9x+ 9

√3y − 63 = 0

Identifying the conic section


In the case we did last time, where B = 0, we identified the casesshape condition if B = 0:

parabola A = 0 or C = 0

AC = 0

ellipse A and C same signs

AC > 0

hyperbola A and C different signs

AC < 0Just like we did for calculating the coefficient of xy after changingcoordinates, we could calculate all the other coefficients after achange in coordinates

0 = A(x)2 +Bxy + C(y)2 +Dx+ Ey + F.If we did, we could also show that the discriminants before andafter the change are the same, i.e.

B2 − 4AC = (B)2 − 4AC.But if we chose θ so that B = 0 (like we’ve been doing), then weget

B2 − 4AC = −4AC.




parabola A = 0 or C = 0 AC = 0ellipse A and C same signs AC > 0

hyperbola A and C different signs AC < 0Just like we did for calculating the coefficient of xy after changingcoordinates, we could calculate all the other coefficients after achange in coordinates

0 = A(x)2 +Bxy + C(y)2 +Dx+ Ey + F.

If we did, we could also show that the discriminants before andafter the change are the same, i.e.


B2 − 4AC = −4AC.







B2 − 4AC = (B)2 − 4AC.

But if we chose θ so that B = 0 (like we’ve been doing), then weget

B2 − 4AC = −4AC.








B2 − 4AC = −4AC.

Putting it together: we start with0 = Ax2 +Bxy + Cy2 +Dx+ Ey + F,

and convert to0 = A(x)2 +Bxy + C(y)2 +Dx+ Ey + F,

usingx = x cos(θ)− y sin(θ), y = x sin(θ) + y cos(θ).

We choose θ such that B = 0, i.e. tan(2θ) = B/(A− C).Then since B2 − 4AC = (B)2 − 4AC, we get

−4AC = B2 − 4AC.

Using our table from before, we can classify the conic sectionaccording to the following table:

shape condition

parabola AC = 0

B2 − 4AC = 0

ellipse AC > 0

B2 − 4AC < 0

hyperbola AC < 0

B2 − 4AC > 0




We choose θ such that B = 0, i.e. tan(2θ) = B/(A− C).

Then since B2 − 4AC = (B)2 − 4AC, we get

−4AC = B2 − 4AC.


shape condition

parabola AC = 0

B2 − 4AC = 0

ellipse AC > 0

B2 − 4AC < 0

hyperbola AC < 0

B2 − 4AC > 0





−4AC = B2 − 4AC.


shape condition

parabola AC = 0

B2 − 4AC = 0

ellipse AC > 0

B2 − 4AC < 0

hyperbola AC < 0

B2 − 4AC > 0





−4AC = B2 − 4AC.


shape condition

parabola AC = 0 B2 − 4AC = 0

ellipse AC > 0 B2 − 4AC < 0

hyperbola AC < 0 B2 − 4AC > 0

You try

Identify the kind of conic section given by each of the followingequations:

1. x2 − xy − y2 − 2 = 0

2. 4x2 − 4xy + 7y2 − 24 = 0

3. 2x2 + 4√3xy + 6y2 − 9x+ 9

√3y − 63 = 0

Graphing 0 = Ax2 +Bxy + Cy2 +Dx+ Ey + F

1. If B 6= 0, change coordinates using x = x cos(θ)− y sin(θ),y = x sin(θ) + y cos(θ), where tan(2θ) = B/(A− C).Tip: Use a triangle and the half-angle formulas to calculatecos(θ) and sin(θ).

2. With your new equation0 = A(x)2 + C(y)2 +Dx+ Ey + F,

get into one of the canonical forms and calculate all theimportant features of the graph.(If B = 0, then A = A, C = C, etc..)

3. Sketch the graph by first sketching the rotated axes and thengraphing theˆcurve on those axes.

4. If needed, convert features of the graph (like vertices, foci,asymptotes, etc.) back into x, y-coordinates using

x = x cos(θ) + y sin(θ) and y = −x sin(θ) + y cos(θ).

You try: Sketch a graph of 4x2 − 4xy + 7y2 − 24 = 0.

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